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It is proved that if A_p is a countable elementary abelian p-group, then: (i) The ring End(A_p) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End(A_p)/I, where I is the ideal of End(A_p) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) The finite topology on End(A_p) is the only second metrizable ring topology on it. Moreover, a characterization of completely simple endomorphism rings of the endomorphism rings of modules over commutative rings is also obtained.
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Let $H$ be a Hopf algebra, $A/B$ be an $H$-Galois extension. Let $D(A)$ and $D(B)$ be the derived categories of right $A$-modules and of right $B$-modules respectively. An object $M^cdotin D(A)$ may be regarded as an object in $D(B)$ via the restriction functor. We discuss the relations of the derived endomorphism rings $E_A(M^cdot)=op_{iinmathbb{Z}}Hom_{D(A)}(M^cdot,M^cdot[i])$ and $E_B(M^cdot)=op_{iinmathbb{Z}}Hom_{D(B)}(M^cdot,M^cdot[i])$. If $H$ is a finite dimensional semisimple Hopf algebra, then $E_A(M^cdot)$ is a graded subalgebra of $E_B(M^cdot)$. In particular, if $M$ is a usual $A$-module, a necessary and sufficient condition for $E_B(M)$ to be an $H^*$-Galois graded extension of $E_A(M)$ is obtained. As an application of the results, we show that the Koszul property is preserved under Hopf Galois graded extensions.
It is proved that localizations of injective $R$-modules of finite Goldie dimension are injective if $R$ is an arithmetical ring satisfying the following condition: for every maximal ideal $P$, $R_P$ is either coherent or not semicoherent. If, in addition, each finitely generated $R$-module has finite Goldie dimension, then localizations of finitely injective $R$-modules are finitely injective too. Moreover, if $R$ is a Prufer domain of finite character, localizations of injective $R$-modules are injective.
Let $A$ and $B$ be rings, $U$ a $(B, A)$-bimodule and $T=left(begin{smallmatrix} A & 0 U & B end{smallmatrix}right)$ be the triangular matrix ring. In this paper, we characterize the Gorenstein homological dimensions of modules over $T$, and discuss when a left $T$-module is strongly Gorenstein projective or strongly Gorenstein injective module.
Let R be a ring (not necessarily commutative). A left R-module is said to be cotorsion if Ext 1 R (G, M) = 0 for any flat R-module G. It is well known that each pure-injective left R-module is cotorsion, but the converse does not hold: for instance, if R is left perfect but not left pure-semisimple then each left R-module is cotorsion but there exist non-pure-injective left modules. The aim of this paper is to describe the class C of commutative rings R for which each cotorsion R-module is pure-injective. It is easy to see that C contains the class of von Neumann regular rings and the one of pure-semisimple rings. We prove that C is strictly contained in the class of locally pure-semisimple rings. We state that a commutative ring R belongs to C if and only if R verifies one of the following conditions: (1) R is coherent and each pure-essential extension of R-modules is essential; (2) R is coherent and each RD-essential extension of R-modules is essential; (3) any R-module M is pure-injective if and only if Ext 1 R (R/A, M) = 0 for each pure ideal A of R (Baers criterion).
Let $T=left( begin{array}{cc} R & M 0 & S end{array} right) $ be a triangular matrix ring with $R$ and $S$ rings and $_RM_S$ an $R$-$S$-bimodule. We describe Gorenstein projective modules over $T$. In particular, we refine a result of Enochs, Cort{e}s-Izurdiaga and Torrecillas [Gorenstein conditions over triangular matrix rings, J. Pure Appl. Algebra 218 (2014), no. 8, 1544-1554]. Also, we consider when the recollement of $mathbb{D}^b(T{text-} Mod)$ restricts to a recollement of its subcategory $mathbb{D}^b(T{text-} Mod)_{fgp}$ consisting of complexes with finite Gorenstein projective dimension. As applications, we obtain recollements of the stable category $underline{T{text-} GProj}$ and recollements of the Gorenstein defect category $mathbb{D}_{def}(T{text-} Mod)$.
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